Assouad dimension

The Assouad dimension of the Sierpiński triangle is equal to its Hausdorff dimension, $\alpha =\frac{\log (3)}{\log (2)}$. In the illustration, we see that for a particular choice of Template:Mvar, Template:Mvar, and Template:Mvar, $$N_r(B_R(x)\cap E)=3=2^{\alpha }={\left(\frac{R}{r}\right)}^{\alpha }.$$ For other choices, the constant Template:Mvar may be greater than 1, but is still bounded.

In mathematics — specifically, in fractal geometry — the Assouad dimension is a definition of fractal dimension for subsets of a metric space. It was introduced by Patrice Assouad in his 1977 PhD thesis and later published in 1979, although the same notion had been studied in 1928 by Georges Bouligand. As well as being used to study fractals, the Assouad dimension has also been used to study quasiconformal mappings and embeddability problems.

Definition

Let $(X,d)$ be a metric space, and let Template:Mvar be a non-empty subset of Template:Mvar. For Template:Math, let $N_r(E)$ denote the least number of metric open balls of radius less than or equal to Template:Mvar with which it is possible to cover the set Template:Mvar. The Assouad dimension of Template:Mvar is defined to be the infimal $\alpha \ge 0$ for which there exist positive constants Template:Mvar and $\rho$ so that, whenever $$0<r<R\le \rho ,$$ the following bound holds: $$\underset{x\in E}{\sup }{N}_r(B_R(x)\cap E)\le C{\left(\frac{R}{r}\right)}^{\alpha }.$$

The intuition underlying this definition is that, for a set Template:Mvar with "ordinary" integer dimension Template:Mvar, the number of small balls of radius Template:Mvar needed to cover the intersection of a larger ball of radius Template:Mvar with Template:Mvar will scale like Template:Math.

Relationships to other notions of dimension

• The Assouad dimension of a metric space is always greater than or equal to its Assouad–Nagata dimension.
• The Assouad dimension of a metric space is always greater than or equal to its upper box dimension, which in turn is greater than or equal to the Hausdorff dimension.
• The Lebesgue covering dimension of a metrizable space Template:Mvar is the minimal Assouad dimension of any metric on Template:Mvar. In particular, for every metrizable space there is a metric for which the Assouad dimension is equal to the Lebesgue covering dimension.

Further reading

• Template:Cite book